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Advances in Mathematics � AI1630

advances in mathematics 128, 306�328 (1997)

Volume Comparison and Its Applicationsin Riemann�Finsler Geometry

Zhongmin Shen*

Department of Mathematical Science, Indiana University�Purdue University at Indianapolis,Indianapolis, Indiana 46202-3216

Received May 9, 1996; accepted December 30, 1996

We introduce a new geometric quantity, the mean covariation for Finsler metrics,and establish a volume comparison theorem. As an application, we obtain someprecompactness and finiteness theorems for Finsler manifolds. � 1997 Academic Press

1. INTRODUCTION

Early work in conjunction with the global comparison geometry of Finslermanifolds was done by L. Auslander [A], who proved that if a completeFinsler n-manifold (M, F ) satisfies RicM�(n&1), then the diameterdiamM�?. In particular, ?1(M) is finite. This is the Bonnet�Myerstheorem if F is Riemannian. Further, he proved that if a complete Finslermanifold (M, F ) has non-positive flag curvature, then the exponentialmap expx : TxM � M is a local C1-diffeomorphism. This is the Cartan�Hadamard theorem if F is Riemannian. We point out that Auslander usedthe Cartan connection and stated the above theorems under slightlystronger conditions on curvatures, because he did not realize that the cur-vature terms in the second variation formula can be simplified to the flagcurvature. See [AZ2, BCh] for simplified formulas. We remark that theflag curvature is independent of a particular choice of connections, andhence is of particular interest in the metric geometry of Finsler manifolds.See [L, AZ1, D1�D3, GKR, Mo1, Mo2, K1, K2, BCh], etc., for otherinteresting work on global Finsler geometry.

In Riemann geometry, the Bishop�Gromov volume comparison theorem[BCr, GLP] plays a very important role in the global differential geometryof Riemann manifolds. One of its applications is the Gromov precompact-ness theorem for the class C(n, *, d ) with respect to the Gromov�Hausdorffdistance dGH , where C(n, *, d) denotes the set of all Riemann manifolds(M, g) satisfying RicM�(n&1)* and diamM�d. For =>0, R�1, consider

article no. AI971630

3060001-8708�97 �25.00Copyright � 1997 by Academic PressAll rights of reproduction in any form reserved.

* E-mail: zshen�math.iupui.edu.


a sub-class C=, R(n, *, d )/C(n, *, d ), which consists of (M, g) satisfying thatany =-ball B(x, =) is contractible in B(x, R=)/M. It can be shown that iftwo manifolds M0 , M1 # C=, R(n, *, d ) are sufficiently close with respect todGH , then they have to be homotopy equivalent. Combining this with thepre-compactness of C(n, *, d ), one can conclude that there are only finitelymany homotopy types in C=, R(n, *, d ) ([P1]) (see also [GPW] for ahomeomorphism version of this). This is a generalized version of Cheeger'sfiniteness theorem [C1, C2] and Grove�Petersen's finiteness theorem[GP]. See also [Y, Z] for some other interesting finiteness theorems. Herewe shall not mention the references on the convergence theory of Riemannmetrics, because we will only focus on the finiteness problem. The readeris referred to a recent paper of P. Petersen [P3] for a beautiful discussionand references on the convergence of Riemannian metrics.

Petersen [P1, P2] studied the finiteness problem for metric spaces ingreat generally. This enables us to find geometric (curvature) conditionsand establish a finiteness theorem for a much larger class of manifoldsequipped with ``smooth'' inner metrics.

First we shall briefly discuss inner metrics which naturally lead to Finslermetrics. Let d be an inner metric on a C� n-manifold M. Suppose itsatisfies the following Lipschitz condition: at each point x # M, there is acoordinate neighborhood .: V/Rn � U/M such that

A&1 |x0&x1 |�d(.(x0), .(x1 ))�A |x0&x1 |, \x0 , x1 # V; (1.1a)

|d(.(x0), .(x0+y )&d(.(x1), .(x1+y )|

�A |x0&x1 | | y |, \x0 , x1 # V, y # Bn(=). (1.1b)

We can prove that (1.1) implies that the following limit exists:

Fd (x, y ) := limr � 0

d(.(x), .(x+ry ))|r|

�0, \x # V, y # Rn (1.2)

Further, F(x, y )=Fd (x, y ) has the following properties:

F(x, y )=|r| F(x, y ), \r # R, y # Rn ; (1.3)

F(x, y )=0 if and only if y=0; (1.4)

|F(x0 , y )&F(x1 , y )|�A |x0&x1 | | y |, \x0 , x1 # V, y # Rn ; (1.5)

F(x, y1+y2)�F(x, y1 )+F(x, y2), \y1 , y2 # Rn. (1.6)

A function F : TM � [0, �) is called a Lipschitz Finsler metric if itsatisfies (1.3)�(1.6) in local coordinates. One can show that if d=dF is theinner metric induced by a Lipschitz Finsler metric F, then d satisfies (1.1)

307RIEMANN�FINSLER GEOMETRY


and reproduces F=Fd by (1.2) ([BM]; see also [V, DCP] for the studyof inner metrics satisfying (1.1a) only). Thus there is an one-to-one corre-spondence between Lipschitz inner metrics and Lipschitz Finsler metrics.

To study the differential geometry of Finsler manifolds, we shall restrictourselves to C� Finsler metrics. A function F : TM � [0, �) is called a C�

Finsler metric if it satisfies (1.3) and the following convexity�regularityconditions:

(1.7) F is smooth on TM"[0];

(1.8) the Hessian ( gij (x, y )), y{0, is positive definite, where

gij (x, y ) :=12

�2F 2

�yi �y j (x, y ).

Note that (1.7), (1.8) imply (1.4), (1.5) and (1.6). The inner metric d=dF

induced by F reproduces F=Fd by (1.2). By the homogeneity of F, onealways has F(x, y )=- gij (x, y ) yiy j . F is called Riemannian if gij (x)=gij (x, y ) are independent of y. F is called locally Minkowskian if gij (x, y )=gij ( y ) are independent of x.

Let Fx denote the restriction of F onto TxM. When F is Riemannian,(TxM, Fx) are all isometric to the Euclidean spaces Rn. But, for generalFinsler metric F, (TxM, Fx) may be not isometric to each other. Thus thegeometry of Finsler manifolds becomes more complicated.

The Finsler metric F induces a familly of inner products gv in TxM by

gv(u, w)=gij (x, y )uiw j, (1.9)

where v=yi (��xi )|x , u=ui (��xi )|x , w=wi (��xi )|x .Take an arbitrary basis [ei ]n

i=1 for TxM. Let Bx(1)=[ y=( yi ):F( yiei )�1]. Put gv

ij=gv(ei , ej ). Define the mean distortion +: TM"[0] �(0, �) by

+(v) :=vol(Bx(1))vol(Bn(1))

- det( gvij ). (1.10)

Clearly, +(v) is independent of a particular choice of [ei ]ni=1 , and

+(*v)=+(v), \*{0.The mean covariation H: TM"[0] � R is defined by

H(v) :=ddt

[ln +(#* v)]| t=0 , (1.11)

where #v is the geodesic with #* v(0)=v. Put H(0)=0. It is easy to see that

H(*v)=*H(v), \* # R.

308 ZHONGMIN SHEN


Roughly speaking, the mean covariation H(v) measures the average rate ofchanges of (TxM, Fx) in the direction v # TxM. We say |HM |�+ if|H(v)|�+F(v), \v # TM"[0].

An important property is that H=0 for Finsler manifolds modeled on asingle Minkowski space. In particular, H=0 for Berwald spaces. LocallyMinkowski spaces and Riemann spaces are all Berwald spaces.

In order to introduce other geometric quantities, one needs (linear) con-nections. In Finsler geometry, there are several important linear connec-tions such as the Berwald connection [Be], the Cartan connection [C],and the Chern connection [Ch1, Ch2, BCh]. See [S] for a new interestinglinear connection and its relationship with others. All of them are uniquelydetermined by two equations (torsion equation and metric equation) andreduce to the common one when F is Riemannian. Except for the Cartanconnection, all of them are torsion-free. But the Cartan connection ismetric-compatible. In a Finsler manifold (M, F ), one cannot get a connec-tion which is torsion-free and metric-compatible, unless F is Riemannian.Using any of these connections, one can define three curvatures R, P, Q(Q=0 for torsion free connections). These curvature terms depend on aparticular choice of linear connections. The flag curvature tensor (definedby R only) is independent of a particular choice of these connections, thatis, the term appears in the second variation of length, thus is of particularinterest to us. We remark that if F is Riemannian, then P=Q=0, and theflag curvature tensor is the Riemannian curvature tensor. The average ofthe flag curvature tensor is the Ricci curvature Ric: TM � R. It has theproperty that Ric(rv)=r2 Ric(v), r>0, v # TM. We say RicM�(n&1)* ifRic(v)�(n&1)*F(v)2, \v # TM. More details will be given in Section 2.

For a constant * # R and +�0, put

V*, +(r) :=Vol(Sn&1(1)) |r

0e+ts*(t)n&1 dt, (1.12)

where s*(t) denotes the unique solution to y"+*y=0 with y(0)=0,y$(0)=1. Note that V*, +(r)=vol(Bn(r))(1+o(r)) as r � 0+. We shall showthat vol(B(x, r))=vol(Bn(r))(1+o(r)) as r � 0+. Thus

limr � 0+

vol(B( p, r))V*, +(r)

=1. (1.13)

The following is our main theorem.

Theorem 1.1. Let (M, F ) be complete Finsler manifold. Suppose that

RicM�(n&1)*, |HM |�+. (1.14)



Then for any 0<r<R,

vol(B(x, R))V*, +(R)

�vol(B(x, r))

V*, +(r). (1.15)

In particular,

vol(B(x, r))�V*, +(r). (1.16)

In Section 6, we shall discuss the case when the equality in (1.15) holds.Given n, *, +�1, let M(n, *, +) denote the class of pointed complete

Finsler n-manifold (M, p, F ) satisfying the bounds (1.14). As a direct conse-quence of Theorem 1.1, we have the following

Corollary 1.2. The class M(n, *, +) is precompact in the pointedGromov�Hausdorff topology.

This corollary follows from Theorem 1.1 and [GLP, P2].A function \: [0, r) � [0, �) is called a contractibility function if it

satisfies (i) $0)=0, (ii) \(=)�=, (iii) \(=) � 0 as = � 0, (iv) \ is non-decreasing. Given a contractibility function \, a metric space X is said tobe LGC($ if for every = # [0, r] and x # X, the ball B(x, =) is contractibleinside B(x, $=)). For a number r>0, if every ball B(x, =) is contractibleinside B(x, =), x # X, 0<=<r, namely, X is LGC($ for $=)== : [0, r) �[0, r), then we say X has contractibility radius c(X )�r. Given n, *,+, \, d, let M(n, *, +, d, $ denote the class of compact LGC(\) Finslern-manifolds satisfying

RicM�(n&1)*, |HM |�+, diamM�d. (1.17)

Corollary 1.3. The class M(n, *, +, d, \) contains only finitely manyhomotopy types.

In [C1, C2], J. Cheeger proved that if a compact Riemannian manifoldMn satisfies the bound 4�KM�*, diamM�d, volM�v, then the injec-tivity radius injM�io(n, *, 4, d, v)>0. Thus M is LGC(\) space for$=)==: [0, io] � [0, �). In [GP], Grove and Petersen proved that if acompact Riemannian manifold Mn satisfies the bounds KM�*, diamM�d,volM�v, then every ball B(x, =) is contractible in B(x, R=) for \=�=o(n, *, d, v) and R=R(n, *, d, v). Thus M is a LGC($ space for$=)=R= : [0, =o] � [0, �). It is an interesting question under what cur-vature bounds is a compact Finsler manifold Mn satisfying the boundsdiamM�d, volM�v a LGC($ space for some \ depending only on thosebounds. This problem will be discussed somewhere else. We remark that

310 ZHONGMIN SHEN


both arguments in [C1] and [GP] were carried out using Toponogov'scomparison theorem. However, we can show that, in a Finsler manifold,Topogonov's triangle comparison theorem does not hold, unless it isRiemannian. In other words, a Finsler manifold is not curved from belowin the sense of Alexandrov [A1, A2] unless it is Riemannian.

By the same argument as in [GLP], one can obtain the following

Corollary 1.4. Let (M, F ) be a complete Finsler n-manifold satisfyingthe bounds (1.17). Then the first Betti number b1(M)�c(n, *, +, d ).

There are many other important theorems in Riemannian geometry arefalse for Finsler manifolds. For example, the Cheeger�Gromoll splittingtheorem [CG] is no longer true, even for flat Finsler manifolds (sayMinkowski spaces). Nevertheless, Milnor's theorem still holds for certainFinsler manifolds.

Corollary 1.5. Let (M, F ) be a complete Finsler n-manifold with

RicM�0, HM=0.

Then any finitely generated subgroup 1 of the fundamental group ?1(M) haspolynomial growth of order �n.

2. PRELIMINARIES

In this section we shall recall some basic facts of Riemann�Finslergeometry. See [M, R, BCh] for more details.

We begin with the simplest Finsler manifolds. Let V be a vector space,and let Fo : V � [0, �) be a function satisfying (i) Fo(*y)=|*| F( y ), (ii) Fo

is C� on V"[0], and (iii) gij ( y ) := 12 (�F 2

o��yi �y j )( y ) is positive definitefor y{0. Fo is called a Minkowski norm on V, and (V, Fo) is called aMinkowski space. Each TxV is naurally identified with V. Thus Fo inducesa Finsler metric F on V.

There are lots of Minkowski spaces, but there is a unique Euclideanspace, up to an isometry. Let (M, F ) be an arbitrary Finsler manifold. Bydefinition, the restriction Fx of F to Tx M is a Minkowski norm in TxM.(TxM, Fx) is called a Minkowski tangent space at x # M. In general,(TxM, Fx) are not linearly isometric to each other. It is possible that on(M, F ), we have infinitely many distinct Minkowski tangent spaces.

There is no notion of angles between two tangent vectors in a Finslermanifold. Nevertheless, the volume form is well-defined [B1]. Let [ei ]n

i=1

be a local basis for TM and ['i]ni=1 be its dual basis for T*M. Put



Bx(1) :=[ y=( yi ) : F( yiei )�1]. Bx(1) is a strictly convex open subsetin Rn. Let Bn(1) denote the standard unit ball in Rn. The volume form dvis defined by

dv=vol(Bn(1))vol(Bx(1))

'1 7 } } } 7 'n, (2.1)

where vol(A) denotes the Euclidean volume of a subset A/Rn. dv is inde-pendent of a particular choice of positive basis [ei ]n

i=1! As usual, thevolume vol(U ) of an open subset U/M is defined by vol(B(x, r))=�U dv.It is easy to see that the r-ball B(x, r) in a Minkowski space has the samevolume of Bn(r) in the Euclidean space Rn. Busemann [B1] proved thatfor any bounded open subset U/M, vol(U )=Hd (U ), where Hd (U )denotes the Hausdorff measure of U with respect to d=dF . This fact mightbe true for Lipschitz inner�Finsler metrics (compare [V]).

In order to define curvatures, it is more convenient to consider thepull-back tangent bundle than the tangent bundle, because our geometricquantities depend on directions.

Let TMo=TM"[0] and let ?*TM denote the pull-back of the tangentbundle TM by ?: TMo � M. Denote vectors in ?*TM by (v; w), v # TMo ,w # T?(v) M. For the sake of simplicity, we denote by �i | v=(v; ��xi |x),v # TxM the natural local basis for ?*TM. The Finsler metric F defines twotensors g and A in ?*TM by

g(�i | v , �j | v)=gij (x, y), A(�i | v , � j | v , �k | v)=12

F(x, y)�gij

�yk (x, y),

where v=yi (��xi )|x . g and A are called the fundamental and Cartantensors, respectively. Note that (?*TM, g) is a Riemannian vector bundle.A trivial fact is that F is Riemannian if and only if A=0.

In Finsler geometry, we study connections and curvatures in (?*TM, g),rather than in (TM, F ). The pull-back tangent bundle ?*TM is a veryspecial vector bundle. It has a unit vector l defined by

lv=1

F(v)(v; v).

It is easy to see that A(X1 , X2 , X3)=0 whenever Xi=l for some i=1, 2, 3.Let [Ei ]n

i=1 be a local frame for ?*TM. Define the dual co-frame[|i ]n

i=1 on TMo by

(v; ?*

(X� ))=|i(X� )Ei , X� # T(TMo).

Put l=l iEi , Aijk=A(Ei , Ej , Ek), gij=g(Ei , Ej ).

312 ZHONGMIN SHEN


We have the following

Theorem 2.1 (Chern). There is a unique set of local 1-forms [|ji ]1�i, j�n

on TMo such that

d|i=| j 7 |ji (2.2)

dgij=gkj|ik+gik |j

k+2Aijk |n+k, (2.3)

where |n+k :=dlk+l l|lk+l kd(ln F ). Further, [|i ; |n+i] is a local co-

frame for T*(TMo).

In a standard local coordinate system (xi ; yi ) in TM, take a naturalbasis [Ei=�i]n

i=1 for ?*TM. We have

|ji=1 i

jk(x, y) dxk

where

1 ijk=#i

jk+1F

gil [AjksN sl &AljsN s

k&Akls N sj ]

N sl =#s

la ya&1F

gsjAjlk#kab yayb (2.4)

#ijk=

12

gil { ��x j glk+

��xk glj&

��x l gjk= .

We call [|ji] the set of local connection forms. It defines a linear connec-

tion { in ?*TM by

{X� Y=[X� Yi+Y j|ji (X� )]Ei ,

X� # T(TMo), Y=YiEi # C�(?*TM). (2.5)

We also get two bundle maps \, +: T(TMo) � ?*TM, defined by

\ :=|i �Ei , + :=F|n+i�Ei (2.6)

Note that the VTM :=ker \ is the vertical tangent bundle of TMo . PutHTM :=ker +. We have the direct composition T(TMo)=HTM�VTM.Tangent vectors in HTM are called horizontal, and tangent vectors in VTMare called vertical. An important fact is that \ |HTM and + |VTM are bundleisomorphisms.

Define the set of local curvature forms 0ji by

0ji :=d|j

i&|jk 7 |k

i.



By (2.2), one can show that 0ji does not have vertical part, that is, one can

write

0ji= 1

2Rjikl|k 7|l+Pj

ikl|k 7 |n+l.

Define the curvature tensors R, P in ?*TM by

R(U, V )W=ukvlw jRjikl Ei , P(U, V)W=ukvlw jPj

iklEi ,

where U=uiEi , V=vi Ei , W=wiEi # ?*TM. A Finsler manifold (M, F ) iscalled a Berwald space if P=0.

Let _=span[u, v]/TxM be a two-dimensional section. The flag cur-vature K(_; v) of the flag [_, v] is defined by

K(_; v) :=g(R(U, V )V, U )

g(V, V) g(U, U )&g(U, V )2 ,

where U=(v; u), V :=(v; v) # ?*TM. When F is a Riemannian, K(_)=K(_; v)is independent of v # _, that is, the sectional curvature in Riemanniangeometry. Further, it is independent of the above-mentioned linear connec-tions. Thus it really does not matter which connection should be used inthe metric geometry of Finsler manifolds.

Fixing a unit vector v # TxM, let [ei ]ni=1 , en=v, be a basis for TxM

such that [(v; ei )]ni=1 is an orthonormal basis for ?*TM. Let _i=

span[ei , v], 1�i�n&1. The Ricci curvature Ric(v) is defined by

Ric(v) := :n&1

i=1

K(_i ; v)F(v)2.

The linear connection { in (2.5) defines the covariant derivative Dvu ofa vector field u on M in the direction v # TxM as follows. Let c be a curvein M with c* (0)=v. Let c=dc�dt be the canonical lift of c in TMo . Letu(t)=u| c(t) and U(t) :=(c; u(t)) # ?*TM. Define Dvu by

(v; Dvu) :={dc�dtU(0). (2.7)

Note that D satisfies all properties of linear connections in TM, except forthe linearity in v, that is, Dv1+v2

u{Dv1u+Dv2

u. Thus D is not a linear con-nection in TM in a usual sense. A vector field u=u(t) along c is calledparallel if Ddc�dt u=0.

A curve #: [0, a] � M is a geodesic if and only if #* is parallel along #, i.e.,

D#* #* =0. (2.8)

In this case, # must be parametrized proportional to arc-length.

314 ZHONGMIN SHEN


The exponential map expx : TxM � M is defined as usual, that is,expx(v)=#v(1), where #v is a geodesic with #v(0)=x and #* (0)=v. TheHopf�Rinow theorem says that if (M, dF ) is complete, then expx is definedon all of TxM for all x # M. This implies that any two points x0 , x1 # Mcan be joined by a minimizing geodesic. From now on, we always assumethat (M, dF ) is complete.

It is a well-known fact that the exponential map expx is C� away fromthe origin in TxM and only C1 at the origin with expx | 0=identity [W].In [B2], Busemann proved that if expx is C2 at the origin for all x # M,then all (TxM, Fx) are isometric to each other. On the other hand, Ichijyo�[I] proved that in a Berwald space, all (TxM, Fx) are isometric to eachother. Finally, Akbar�Zadeh [AZ3] proved that expx is C� all over TxMfor all x # M, if and only if the Finsler metric is Berwald.

Using the exponential map expx , one can easily show that d 2x=d(x, } )2

is C� near x and C1 at x. We remind the reader that (Fx)2 is only C1 atthe origin, although expx is C� in a Berwald space.

Proposition 2.2. If d 2x is C2 at x, then F is Riemannian at x.

Proof. Let .: V/Rn � U/M be a local coordinate system at x with.(0)=x. Let h(z)=d 2

x b .(z), z # V. Note that h(z)�h(0)=0, z # V. Byassumption h is C2 in V. Thus

h(z)=12

�2h�zi �z j (0) ziz j+o( |z| ).

As we have pointed in Section 1, d=dF reproduces F=Fd by (1.2), that is,

F(x, y)2= limr � 0

h(ry )2

r2 =12

�2h�zi �z j (0) yiy j.

By definition, F is Riemannian at x. K

The cut-value tv of a vector v # TxM is defined to be the largest numberr>0 such that #v is minimizing on [0, r]. Let Ix :=[v # TxM, F(v)=1].The map v # Ix � tv # [0, �) is continuous. The cut-locus Cx=[expp tvv: v # Ix] has zero Hausdorff measure in M. The injectivityradius injx at x is defined by injx=infv # Ix

tv . Let 0x :=M"Cx , andOx=[(t, x): t<tv]. Then expx : Ox � 0x is a diffeomorphism (C1 at theorigin).

For v # Tx M, define Rv : TxM � TxM by

Rv(u)=R(U, V )V, (2.9)

where U=(v : u), V=(v; V ) # ?*TM.



Fix v # Tx M, and let #v be a normal geodesic from x with #* v(0)=v.Along #v , we have a family of inner products gt=g#* v (t) in T#v (t) M (see(1.9)) and flag curvatures Rt=R#* v (t) : T#v (t) M � T#u (t)M. For the sake ofsimplicity, we shall denote Dt=D#* v

, if no confusion is caused. (2.3) impliesthat for any vector fields u=u(t), w=w(t) along #v ,

ddt

( gt(u(t), w(t))=gt(Dtu(t), w(t))+gt(u(t), Dtv(t)). (2.10)

A vector field J=J(t) along #v is called a Jacoby field if it satisfies

DtDt J(t)+RtJ(t)=0. (2.11)

Lemma 2.3. A vector field Ju along #v with Ju(0)=0 and DtJu(0)=u isa Jacobi field if and only if

Ju(t)=d(expx) |tv (tu). (2.12)

In particular, a vector field J given by (2.12) is smooth along #v .

We shall always denote by #* v the geodesic with #* v(0)=v # TxM and byJu the Jacobi field along #* v defined by (2.12).

Lemma 2.4 (Gauss Lemma). For every Jacobi field Ju , Ju(t) is per-pendicular to #v(t) for all t with respect to gt, if and only if u is perpendicularto v with respect to gv.

From (2.12), we can see that expx is singular at rv # TxM if and only ifthere is 0{u # TxM, such that the Jacobi field Ju satisfies Ju(r)=0. In thiscase we call #v(r) a conjugate point with respect to x. By the standard argu-ment, one can show that if the flag curvature K(_; v)�*, then there areno conjugate points on # | [0, r) , for r=?�- * (=� if *�0). This is theso-called Cartan�Hadamard theorem in Finsler geometry [A].

The index lemma is still true. For a vector field W=W(t) along #v withW(0)=0, define

I(W, W )=|r

0[ gt(Dt W(t), Dt W(t))&gt(RtW(t), W(t))] dt.

Lemma 2.5 (Index Lemma). Suppose that #v does not contain conjugatepoints on [0, r]. Let J be a Jacobi field along #v with J(0)=0. Then for anyvector field W along # with W(0)=0 and W(r)=J(r),

I(J, J )�I(W, W ),

the equality holds if and only if W=J.

316 ZHONGMIN SHEN


For a Jacobi field Ju , we have

I(Ju , Ju)=gr(Dt Ju(r), Ju(r)).

By a standard argument and the index lemma, one can easily prove Bonnetand Myers' theorem for Finsler manifolds ([A]).

A Finsler space (M, F ) is said to be modeled on a single Minkowski spaceif for every geodesic #, the parallel translation Pt0 , t1

: (T#(t0)M, F#(t0)) �(T#(t1)M, F#(t1)) is an isometry for all t0 , t1 . In this case, all (TxM, Fx) arelinearly isometric to each other. The class of such manifolds contains allRiemann manifolds and locally minkowski manifolds. Define F* : TRn �[0, �) by

F* \ yi ��xi } x+=� :

n

i=1

( yi )2+*� :n

i=1

( yi )4 , (2.13)

where *: Rn � [0, �) is a C� function. Clearly, (Rn, F*) in (2.13) is notmodeled on a single Minkowski space, if *: Rn � [0, �) is not constant.See [AIM] for more interesting Finsler metrics from physics.

Proposition 2.6. If (M, F ) is modeled on a single Minkowski space,then H=0.

Proof. By definition, if u=u(t) is a parallel vector field along a geodesic#v , then F(u(t))=constant. Let [ei (t)]n

i=1 be a parallel basis for T#v (t) Mwith ei (0)=ei , 1�i�n. We have

B#v (t)(1) :=[( yi ): F( yiei(t))�1]=Bx(1).

We also have det( gt(ei (t), ej (t)))=det( gv(ei , ej )). Thus +(#* v(t))=constant.This implies H(v)=0. K

Proposition 2.7 ([I]). Any Berwald space is modeled on a singleMinkowski space.

3. THE SINGULAR RIEMANN METRICS gx AND gx

The Finsler metric F induces a singular Riemann metric gx on Tx M"[0]by

gx(u, w) :=gv(u, w), \u, w # Ttv(TxM)tTxM, v # TxM. (3.1)



Let Ix=[v # TxM: F(v)=1]. Let g* x be the induced Riemann metric of gx

on Ix/TxM. Regard TxM as the cone C(Ix) over Ix . By the homogeneityof F, we have

gx=dt2� t2g* x. (3.2)

Let 'x denote the volume form of gx on TxM"[0]=C(Ix)"[o]. Let[ei ]n

i=1 be a basis for T0(TxM)tTx M. Extend it to a global basis forT(Tx M). Let [%i]n

i=1 be the dual basis for T*(TxM). We have

'x|tv=- det( gx(ei , ej )) |tv %1 7 } } } 7%n

=- det( gv(ei , ej )) %1 7 } } } 7 %n. (3.3)

Let '* x denote the volume form of g* x on Ix . By (3.2) we have

'x|tv=tn&1'* x | v 7 dt, \(t, v) # C(Ix). (3.4)

Define the density 3x at x # M by

3x=vol(Ix , '* x)

vol(Sn&1(1)). (3.5)

When F is Riemannian, 3x=1, \x # M. In general, 3x does not have to beconstant unless F is a weak Landsberg space (see [BS]). It is still an openproblem whether or not there are two universal constants 0<an<bn ,n=dim M such that

an�3x�bn , \x # M. (3.6)

If the norm of the Cartan tensor A is small enough, say &A&�- 3�10, then(3.6) holds for some an , bn .

Define a smooth Riemannian metric gx inside 0x"[x] by

gx | #v (t)=g#* v (t) (3.7)

(see (1.9)). Put

g~ x :=(expx)* gx. (3.8)

In general, gx is singular at the origin, unless F is Riemannian. Regard expx

as a map C(Ix) � M. By the Gauss Lemma, we have

g~ x=dt2�ht , (3.9)

where ht is a family of Riemannian metrics on Ix .

318 ZHONGMIN SHEN


Lemma 3.1. The metric g~ x satisfies

1t2 ht � g* x,

12t

�ht

�t� g* x. (3.10)

Proof. Let u, w # Tv(IxM)/Ttv(TxM)tTxM. Let gt=g#* (t). Clearly,

gt(u(t), w(t)) � g* x(u, w), (3.11)

where u(t), w(t) are parallel vector fields along # with u(0)=u, w(0)=w.For u # TxM, let Ju(t)=d(expx) |tv (tu). We have

ht(u, w)=gx(d(expx) |tv (tu), d(expx) |tv (tw))=gt(Ju(t), Jw(t)). (3.12)

Thus by (2.10)

�ht

�t(u, w)=gt(DtJu(t), Jw(t))+gt(Ju(t), DtJw(t)). (3.13)

Put J� u=(1�t) Ju(t), for t{0. Since J is smooth along # and J(0)=0,then J� u :=(1�t)Ju(t) converges, as t � 0. Put J� (0)=limt � 0 J� u(t). Then J� u issmooth. Thus Dt J� u is bounded. Observe that Dt Ju(t)=J� u(t)+tDt J� u(t). Wecan conclude that

1t

Ju(t) � DtJu(0)=u. (3.14)

Now (3.10) follows from (3.11)�(3.13). K

The following proposition is important in computing the flag curvature.

Proposition 3.2. (i) Let Dx be the Levi�Civita connection of gx in 0x .The along any normal geodesic #v , v # Ix ,

Dxt =Dt . (3.15)

(ii) Let u # Tv(Ix), and let _t=span[d(expx) |tv (u), #* v(t)]/T#* v (t)M.Let Kx(_t) denote the sectional curvature of gx. Then the flag curvatureK(_t , #* v(t)) of F satisfies

K(_t ; #* v(t))=Kx(_t)=&

12

�2

�t2 (ht)uu+14

��t

(ht)u: (ht):; �

�t(ht)+;

(ht)uu, (3.16)

where [e:]n&1:=1 is a basis for Tv(Ix), (ht)uu=ht(u, u), (ht)u:=ht(u, e:), and

(ht):;=ht(e: , e;).



Proof. Let (t, x� ) be the conical (or polar) coordinate system in0x=expx(Ox), regarding TxM as C(Ix). Let (t, x� ; s, y� ) be the standardcoordinate system in T0x. By the Gauss lemma or (3.9), we have

\10

0( gx):; (t, x� )+=\1

00

g:;(t, x� ; 1, 0)+=\10

0(ht):; (t, x� )+ .

Put Dx��t(��x:)=#;

t:(t, x� )(��x;). We have

# ;t:(t, x� )=

12

(ht);& (t, x� )

��t

(ht):&; (t, x� ).

We can put D(��t)(��x:)=1 ;t:(t, x� ; 1, 0)(��x;). By (2.4),

1 ;t:(t, x� ; 1, 0)=# ;

t:(t, x� )&(ht);& (t, x� ) A:&{(t, x� ; 1, 0) #{

tt(t, x� ).

An easy computation yields that #{tt(t, x� )=0. Thus

1 ;t:(t, x� ; 1, 0)=# ;

t:(t, x� ).

This implies (3.15).Let Rx denote the Riemann curvature tensor of gx on 0x . Note that

J:(t) :=d expx | (t, x� ) (t(��x:)=t(��x�)| (t, x� ) is a Jacobi field of both F andgx. By the Jacobi equation (2.11), one obtains

(Rx)t ��x:=&

1t

Dx��tD

x��t J:(t)=&

1t

D��tD��t J:(t)=Rt ��x: .

By an easy computation, one obtain that for _t=span[��t, ��x:],

Kx(_t)=&

��t

[ gt(D��tJ:(t), J:(t))]&gt(D��tJ:(t), D��t J: (t))

( gt)(J:(t), J:(t))

=&

12

�2

�t2 (ht)::+14

��t

(ht):; (ht);& �

�t(ht)&:

(ht)::. K

See [LS] for further discussions on Riemann metrics with conicalsingularities.

320 ZHONGMIN SHEN


4. THE VOLUME FORMS OF gx and gx

Let v # Ix . Let 'x denote the volume form of gx and put

'~ x=(expx)* 'x. (4.1)

Note that '~ x is the volume form of g~ x. Define 3(v, t), t<tv , by

'~ x|tv=3(v, t) t&(n&1) 'x

|tv .

Let #v(t)=expx(tv), t�0, and [e:]n&1:=1 be an orthonormal basis for

v=/TxM with respect to gv. Extend [e:]n&1:=1 to a global set for T(TxM).

Let J:(t)=d(expx) |tv (te:). We have

3(v, t) :=�det(ht(e: , e;))det( g* x(e: , e;))

=�det( gt[J:(t), J;(t)])det( gv(e: , ;))

, t<tv . (4.2)

Clearly, 3(v, t) is independent of a particular choice of [e:]n&1:=1. Lemma 3.1

implies

limt � 0+

3(v, t)s*(t)n&1=1. (4.3)

Lemma 4.1. For t<tv , the function t � 3(v, t)�s*(t)n&1 is monotonedecreasing. In particular,

3(v, t)�s*(t)n&1. (4.4)

Proof. For r<tv , one can choose [e:]n&1:=1 such that gr(J:(r), J;(r))=0,

:{;. Note that Jv(t)=t#* v(t) and is orthogonal to J:(t) for all t>0,:=1, ..., n&1. Let e:(t) be the parallel vector fields along # such thate:(0)=e: and e:(r)=J:(r)�gr(J:(r), J:(r))1�2.

Let W:(t)=(s*(t)�s*(r)) e:(t). By the index lemma and (2.10),

3$(v, r)3(v, r)

= :n&1

:=1

gr(Dt J:(r), J:(r))gr(J:(r), J:(r))

� :n&1

:=1

I(W: , W:)

=1

s*(r)2 |r

o[(n&1) s$*(t)2&Ric(#* (t)) s*(t)2] dt

�1

s*(r)2 |r

0[(n&1) s$*(t)2&(n&1)*s*(t)2] dt

=(n&1)s$*(r)s*(r)

.



This implies

ddr

ln3(v, r)

s*(r)n&1�0.

Therefore 3(v, r)�s*(r)n&1 is monotone decreasing in r # (0, tv). Thistogether with (4.3) implies (4.4). K

Put 3(v, t)=0 for t�tv . The pointed volume of a metric ball B(x, r) isdefined by

Vx(r) :=vol(B(x, r), gx).

Let V*(r)=V*, 0(r). It is easy to see that

Vx(r)=|r

0 \|Ix

3(v, t)+* x+ dt. (4.5)

By the standard argument, one can show that r � Vx(r)�V*(r) is decreasing.By (4.3) we have

limt � 0+

Vx(r)V*(r)

=3x .

Therefore we have

Theorem 4.2. Suppose that (M, F ) satisfies RicM�(n&1)*. Then forevery x # M, the function r � Vx(r)�V*(r) is monotone decreasing. Inparticular,

Vx(r)�3x V*(r).

5. PROOF OF THEOREM 1.1

Fix a basis [ei]ni=1 for T0(Tx M)tTxM. Extend [ei ]n

i=1 to be a globalbasis for T(Tx M). Let [%i ]n

i=1 be a dual basis for T*(TxM).Let #v : [0, �) � M be a normal geodesic with #* v(0)=v # TxM. Let Ji(t)=

d(expx) |tv (tei ) and J� i (t)=d(expx) |tv ei . Put B� t=[( yi ) : F( yiJ� i (t))�1].Let ['~ i ]n

i=1 be the co-frame dual to [J� i (t)]ni=1 . The volume form dv has

the form

dv |expv (tv)=vol(Bn(1))vol(B� t(1))

'~ 1 7 } } } 7 '~ n.

322 ZHONGMIN SHEN


We have

%i|tv=(expx)*|tv '~ i.

Thus

(expx)*|tv dv=vol(Bn(1))vol(B� t(1))

%1 7 } } } 7%n.

Since the mean distortion is independent of a particular choice of [ei]ni=1 ,

we have

+(#* v(t))=vol(B� t(1)) - det( gt(J� i (t), J� j (t))

vol(Bn(1)).

Thus

(expx)*|tv dv=1

+(#* (t))- det( gt(Ji (t), Jj (t))) t&n%1 7 } } } 7%n. (5.1)

Choose [ei]ni=1 with en=v such that e: , 1�:�n&1, are orthogonal to

v with respect to gv. By the Gauss lemma, all J:(t), 1�:�n&1, areorthogonal to Jn(t)=t#* v(t). Therefore

(expx)*|tv dv=1

+(#* (t))- det( gt(J:(t), J;(t))) t&(n&1)%1 7 } } } 7 %n

=3(v, t)+(#* v(t))

t&(n&1)- det( gv(e: , e;)) %1 7 } } } 7 %n

=3(v, t)+(#* v(t))

t&n(n&1)'x

=3(v, t)+(#* v(t))

'* x 7 dt.

Here '* x is the volume form of (IxM, g* x).To prove Theorem 1.1, one needs the following

Lemma 5.1. Suppose that |HM |�+. Then the function t � e&+t�+(#* v(t))is monotone decreasing.



Proof. By the definition of H, we have

H(#* (t))=ddt

[ln +(#* v(t))]�&+=ddt

[ln e&+t]. (5.2)

This implies that t � (e&+t�+(#* v(t))) is monotone decreasing. K

Let

_x(r)=|Ix

3(v, r)+(#* v(r))

'* x, _*, +(r)=vol(Sn&1(1))e+rs*(r)n&1. (5.3)

We have

vol(B(x, r))=|r

0_x(t) dt, V*, +(r)=|

r

0_*, +(t) dt.

By Lemma 4.1 and Lemma 5.1, we conclude that h(r) :=_x(r)�_*, +(r) ismonotone decreasing. By the standard argument [G], the function

h� (r) :=�r

0 _x(t) dt�r

0 _*, +(t) dt

is also monotone decreasing.Next we are going to prove that limr � 0+ h� (r)=1.

Lemma 5.2. For every x # M,

lim= � 0+

vol(B(x, =))V*, +(=)

=1. (5.4)

Proof. The proof is standard. Take the normal coordinates (xi ) at x.Clearly, x$ � vol(Bx$(r)) is continuous. For small =>0, the =-ball B(x, =)/Mis mapped onto Bx(=)/Rn by exp&1

x . Thus as = � 0+,

vol(B(x, =))vol(Bn(=))

=1

vol(Bn(=)) |Bx (=)

vol(Bn(1))vol(Bx$(1))

dx$=vol(Bx(=))vol(Bx $=

(=))� 1. K

Lemma 5.2 implies that

|Ix

1+(v)

'* x=1.

Suppose the mean distortion satisfies

+&1�+(v)�+ \v # TM. (5.5)

324 ZHONGMIN SHEN


By (5.3), we have

+&1Vx(r)�vol(B(x, r))�+Vx(r).

We obtain the following

Theorem 5.3. Let (M, F ) be a complete Finsler n-manifold satisfyingRicM�(n&1)* and (5.5). Then for all r<R,

vol(B(x, R))V*(R)

�+2 vol(B(x, r))V*(r)

.

In particular,

vol(B(x, R))�+2V*(R).

This theorem is sharp for Riemann manifolds, since +(v)=1, but it is notsharp for Finsler manifolds (say Minkowski spaces). Nevertheless, theprecompactness and finiteness theorems can be established for +(v) insteadof H(v).

6. THE RIGIDITY PROBLEM

In this section we shall study the case when

vol(B(x, r))V*, +(r)

=1. (6.1)

By the proof of Theorem 1.1, one can see that injx�r. Further,

3(v, t)=s*(t)n&1, 0�t�r. (6.2)

and

H(#* v(t))=+, 0�t�r. (6.3)

We have H(v)=H(&v), \v # Ix , since (6.3) holds for all v # Ix and t=0.On the other hand, by the definition of H(v), it is easy to see thatH(&v)=&H(v), \v # TxM. Thus +=0 and

H(#* v(t))=0, 0�t�r. (6.4)

Lemma 4.1 and (6.2) imply that any Jacobi field Ju(t) along #v has the form

Ju(t)=s*(t)u(t), 0�t�r, (6.5)



where u=u(t) is a parallel vector field along #v . By the Jacobi field equa-tion, one has

Rtu(t)=u(t). (6.6)

This means the flag curvature for the flag [_t , #* v(t)] is constant 1, where_t=span[#* v(t), u]. We remark that this does not say the flag curvature at#v(t) is constant. We have

Proposition 6.1. If Ju satisfies (6.5) for all u # TxM, then

g~ x|tv=(expx)* gx

|tv=dt2�s*(t)2 g* x.

Proof. By (3.9),

g~ x|tv=(expx)* gx

|tv=dt2�ht

where ht is a family of Riemann metrics on Ix . By Proposition 3.2 and(6.6), for any basis [e:]n&1

:=1 for Tv(Ix),

(ht)uu=&12

�2

�t2 (ht)uu+14

��t

(ht)u: (ht):; �

�t(ht)u; .

Lemma 3.1 says that ht satisfies the initial condition (3.10). By the rigiditytheorem in [LSY], one can conclude that ht=s*(t)2 g* x. K

ACKNOWLEDGMENTS

The author thanks D. Bao for many helpful discussions and S. S. Chern for his greatencouragement.

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328 ZHONGMIN SHEN

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